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________________ 0Ara0 esa0 pAMDeya bhAratIya gaNita kA saMkSipta itihAsa gaNita ko bhArata meM prArambha se hI bahuta mahatvapUrNa viSaya mAnA jAtA rahA hai| 'vedAMga jyotiSa' (1000 I0 pU0) meM gaNita kI mahattA para prakAza DAlate hue likhA gayA hai : yathA zikhA mayUrANAM, nAgANAM maNayo yathA tadvavedAMga zAstrANAM, gaNitaM mUrdhni vrtte| arthAt jisa prakAra mayUroM kI zikhAeM aura sarpo kI maNiyAM zarIra meM sabase Upara mastaka para virAjamAna haiM usI prakAra vedoM ke saba aMgoM tathA zAstroM meM gaNita ziromaNi hai| bhAratIya gaNita ke itihAsa kA zubhArambha Rgaveda se hotA haivaidika kAla (1000 I0 pUrva taka): vedoM meM saMkhyAoM aura dArzanika praNAlI kA spaSTa ullekha milatA hai| Rgaveda kI eka RcA hai : dvAdaza pradhayazya kramekaM triNi nabhyAmika utacciketa ___ tasmintsAmakaM trizatA na zaMkavo'rpita SaSThirna calacalAsa / isameM dvAdaza arthAt bAraha, triNi arthAt tIna, trizati arthAt tIna sau, SaSThi arthAt sATha saMkhyAoM kA prayoga dAzamika praNAlI ke jJAna kA spaSTa udAharaNa hai| ___ isa kAla meM 'zUnya' (zero) aura "dAzamika sthAna mAna" paddhati kA AviSkAra gaNita ke kSetra meM bhArata kI abhUtapUrva dena hai| yaha nizcita rUpa se jJAta nahIM hai ki zUnya kA AviSkAra kaba aura kisane kiyA, kintu isakA prayoga vaidika kAla se hotA rahA hai| 'zUnya' aura 'dAzamika sthAna mAna' kI paddhati Ajakala sampUrNa vizva meM pracalita hai| maharSi vedavyAsa dvArA praNIta nArada viSNu purANa meM triskandha jyotiSa ke varNana prasaMga meM gaNita viSaya kA pratipAdana kiyA gayA hai, jisameM (10deg), daza, zata, sahastra, ayuta (dasa hajAra), lakSa (lAkha), koTi (karor3a), arbuda (dasa karor3a), abja (araba), kharba (dasa araba), mahApadya (dasa kharaba), zaMku (nIla), jaladhi (dazanIla), antya (padya), madhya (dasa padya), parArdha (zaMkha jo 10" ke mAna ke barAbAra hai) ityAdi saMkhyAoM ke bAre meM batAyA gayA hai ki ye saMkhyAeM uttarottara dasa gunI haiN| itanA hI nahIM, isameM gaNita kI aneka saMkriyAoM- yoga, vyavakalana, guNA evaM bhAga, bhinna, varga, vargamUla, ghana, ghanamUla, trairAzika vyavahAra Adi kA vizada varNana hai| aMkoM ko likhane kI dAzamika sthAna mAna paddhati bhArata se araba gayI aura araba se pazcimI dezoM meM phuNcii| araba ke loga 1 se 9 taka ke aMkoM ko 'ilma hindasA' kahate haiM aura pazcimI dezoM meM (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ko Hindu Arabic numerals kahA jAtA hai| uttara vaidika kAla (1000 I0 pU0 se 500 I0 pU0 taka): 1: zulva evaM vedAMga jyotiSa kAla : vibhinna prakAra kI vediyoM aura una para upakhaMDoM ko sahI-sahI nApakara banAne ke prazna ko lekara isa kAla meM rekhA gaNita ke sUtroM kA vikAsa evaM vistAra kiyA gayA jo 'zulva sUtroM ke rUpa meM upalabdha haiN| zulva usa rajju (rassI) ko kahate haiM jo yajJa kI vedI banAne ke lie mApa ke kAma AtI thii| dUrI mApane yA pRthvI para vRtta khIMcane meM zulva kA prayoga hotA thaa| zulva kA artha hai rajju yA rssii| ata: vaha gaNita jo zulva kI sahAyatA lekara vikasita kiyA gayA, use zulva vijJAna yA zulva gaNita kA nAma diyA gyaa| zulva kA paryAyavAcI raju hone ke kAraNa ise raju gaNita bhI kahA gayA jo Age calakara rekhAgaNita meM pariNata ho gyaa| vibhinna prakAra kI yajJa vediyoM ke nirmANa meM rajju kI sahAyatA se pRthvI para abhISTa dUriyAM mApane ke atirikta kRSi yogya bhUmi kI mApa bhI kI jAtI thI, isIlie isakI sahAyatA se vikasita gaNita kA nAma kSetramiti, jyAmiti tathA bhUmiti bhI par3a gyaa| kSetra, jyA, bhU kA eka hI artha hai bhUmi tathA miti kA artha hai maapn| ___ jyAmiti ko grIka bhASA meM jyomITrI kahA jAtA hai| aMgrejI bhASA meM bhI Geometry yathAvat prayukta hotA hai| kucha logoM kA vicAra hai ki "jyomITrI" jyAmiti kA aprabhaMza hai| jyAmiti kA mahatva batAte hue eka prAcIna jaina grantha meM kahA gayA hai- "jyAmiti gaNita kA kamala hai aura zeSa saba tuccha hai|" zulva kAla kI pramukha upalabdhiyoM meM se eka hai samakoNa tribhuja kA prameya arthAt "karNa para banA varga zeSa do bhujAoM para bane vargoM ke yoga ke barAbara hotA hai|" yaha prameya pAithAgorasa se kaI zatAbdiyoM hIraka jayantI smArikA adhyApaka khaNDa/1
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________________ pUrva bhArata meM vyApaka rUpa se pracalita thaa| yahI nahIM zulva kAla se pahale hI isa tathya kI jAnakArarI hamAre deza ke vidvAnoM ko thI kyoMki taittarIya saMhitA (3000 I0pU0) meM eka tathya 392=362+152 diyA huA zulva kAla ke mahAn gaNitajJa, zulva sUtra ke racayitA bodhAyana ne likhA hai- "dIrgha caturazrasyAkSaNyA rajjuH pArzvamAnI tiryagamAnI ca yat pRthagbhUte kurutastadubhayaM karoti" __ arthAt dIrgha caturazra (Ayata) kI tiryaMgamAnI aura pArzvamAnI (lamba aura AdhAra) bhujAeM jo do varga banAtI haiM, unakA yoga akele karNa para bane varga ke barAbara hotA hai| pAithAgorasa (540 I0pU0) ne bodhAyana se lagabhaga 450 varSa bAda isa tathya ko pratipAdita kiyaa| ata: isa prayoga ko pAithAgorasa ke sthAna para 'bodhAyana prameya' kahanA susaMgata hogaa| bodhAyana ne do vargoM ke yoga aura antara ke barAbara varga banAne kI vidhi dI hai aura karaNIgata saMkhyA kA mAna dazamalava ke pAMca sthAnoM taka nikAlane ke lie "sUtra" bhI batAyA hai jisake dvArA... 2 = 1 + 1/3 + 1/3.4 - 1/3.4.34 bodhAyana ne anya karaNIgata saMkhyAoM ke bhI mAna diye haiN| prAcIna bhAratIya jyAmitijJa apane dvArA anveSita sAdhyoM kI upapatti nahIM dete the| ve kevala sUtra likhate the jo yathAsaMbhava saMkSipta rUpa meM hote the, jisake kAraNa unameM yatra-tatra aspaSTatA bhI A jAtI thii| yaha saMkSiptatA hindU jAti ke svabhAva ke anurUpa thI jo ki unakI anya prAcIna kRtiyoM meM bhI parilakSita hotI hai| isI kAla meM jyotiSa kA bhI vikAsa huaa| (kAlagaNanA) samaya, nakSatroM kI sthiti aura gati kI gaNanA ke lie jyotiSa gaNita kA vikAsa huaa| vedAMga jyotiSa (1000 I0 pU0) ke adhyayana se jJAta hotA hai ki usa samaya jyotiSiyoM ko yoga, guNA, bhAga Adi kA jJAna thaa| jaise ___ tithimekAdazAmyastA prvbhaaNshsmnvitaam| vibhajya bhasamUhena tithi nksstrmaadishet|| arthAt tithi ko / / se guNA kareM, usameM parva ke bhAMza ko jor3eM aura phira nakSatra saMkhyA se bhAga deN| isa prakAra tithi ke nakSatra ko btaaveN| sUrya prajJapti kAla : __ jaina sAhitya meM tatkAlIna gaNita kA vistRta vivaraNa upalabdha hai| vAstava meM jitane vistAra se aura spaSTa rUpa se gaNitIya siddhAntoM kA vivecana jaina sAhitya meM kiyA gayA hai, vaha jaina darzana kA veda ke rahasyavAda ke sthAna para jJAna ko sAmAnya logoM kI bhASA aura stara taka pahuMcAne kI pravRtti kA spaSTa dyotaka hai| isa kAla kI pramukha kRtiyAM 'sUryaprajJapti' tathA 'candraprajJapti' (500 I0pU0) jaina dharma ke prasiddha dharma grantha haiN| inameM gaNitAnuyoga kA varNana hai| sUrya prajJapti meM dIrghavRtta kA spaSTa ullekha milatA hai jisakA artha dIrgha (Ayata) para banA parivRtta hai jise parimaNDala nAma se jAnA jAtA thaa| spaSTa hai ki bhAratIyoM ko dIrgha vRtta kA jJAna minamaiksa (350 I0pU0) se lagabhaga 150 varSa pUrva ho cukA thaa| yaha itihAsa jJAta na hone ke kAraNa pazcimI deza minamaiksa ko hI dIrghavRtta kA AviSkAraka mAnate haiN| ullekhanIya hai ki bhagavatI sUtra (300 I0pU0) meM bhI parimaNDala zabda dIrghavRtta ke lie hI prayukta kiyA gayA hai jisake do prakAra bhI batAye gaye haiM- 1 : prataraparimaNDala tathA 2 : ghnprimnnddl| gaNita evaM jyotiSa ke vikAsa meM jainAcAryoM kA zlAghanIya yogadAna rahA hai| ina AcAryoM ne apane granthoM meM gaNita ke aneka adhyayana tatvoM kA mImAMsAtmaka vivecana rocaka DhaMga se udAharaNa sahita prastuta kiyA hai| unhoMne saMkhyA - lekhana paddhati, bhinna trairAzika vyavahAra tathA mizrAnupAta, bIjagaNitIya samIkaraNa evaM inake anuprayoga vividha zreNiyAM, kramacaya, saMcaya, ghAtAMka evaM laghugaNaka ke niyama, samuccaya siddhAnta Adi aneka viSayoM para vizada prakAza DAlA hai| samuccaya siddhAMta ke antargata parimita aparimita, ekala samuccayoM ke aneka udAharaNa diye gaye haiN| laghugaNaka ke lie inhoMne arddhaccheda, trikaccheda, caturthaccheda zabdoM kA prayoga kiyA hai jinakA artha kramaza: log2, log3, log4 hai| jAna nepiyara (1550-1617 I0) ke bahuta pahale laghugaNaka kA AviSkAra evaM vistRta anuprayoga bhArata meM ho cukA thA, jo eka sArvajanIna satya hai| pUrva madhyakAla (500 I0pU0 se 400 I0pU0) taka: isa kAla meM likhI gaI pustakoM "vakSAlI gaNita", "sUrya siddhAMta" aura "gaNitAnuyoga" ke kucha pannoM ko chor3akara zeSa kRtiyAM kAla kavalita ho gayI, kintu ina pannoM se aura madhya yuga ke Arya bhaTTa, brahmagupta Adi ke upalabdha sAhitya se yaha niSkarSa nikalatA hai ki isa kAla meM bhI gaNita kA paryApta vikAsa huA thaa| sthAnAMga sUtra, bhagavatI sUtra aura anuyogadvAra sUtra isa yuga ke pramukha grantha haiN| inake atirikta jaina dArzanika umAsvAti (135 I0pU0) kI kRti "tatvArthAdhigama sUtra bhASya" evaM AcArya yativRSabha (176 I0 ke AsapAsa) kI kRti "tiloyapaNNatI" bhI isa kAla ke prasiddha jaina grantha haiN| vakSAlI gaNita meM aMkagaNita kI mUla saMkriyAeM, dAzamika aMkalekhana paddhati para likhI huI saMkhyAeM, bhinna parikarma, varga, ghana, trairAzika vyavahAra, iSTa karma, byAja, rIti, kraya-vikraya saMbaMdhI prazna, sammizraNa saMbaMdhI prazna, Adi kA vistRta vivaraNa hai| praznoM ke uttara ke parIkSaNa ke niyama bhI die gaye haiN| vakSAlI gaNita isa bAta kA pramANa hai ki bhArata meM 300 I0pU0 bhI vartamAna aMkagaNitIya vidhiyoM kA vyApaka prayoga ho rahA thaa| sthAnAMga sUtra meM 5 prakAra ke ananta kI tathA anuprayoga meM 4 prakAra ke pramApa (Measure) kI bAta kahI gaI hai| isa grantha meM kramacaya (Permutation) evaM saMcaya (Combination) kA bhI varNana hai, jinako kramaza: bhaMga aura vikalpa ke rUpa meM jAnA gayA hai| ullekhanIya hai ki bhagavatI sUtra meM na (n) prakAroM meM se 1-1, 2-2, prakAroM ko eka sAtha lene se jo saMcaya (Combination) banate haiM, unheM ekaka, dvika saMyoga Adi kahA hai aura unakA mAna na, na (na-1)/2, Adi batAyA gayA hai jo Aja bhI pracalita hai| - sUrya siddhAMta meM vartamAna trikoNamiti kA vistRta varNana milatA hai| hIraka jayantI smArikA adhyApaka khaNDa/2
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________________ jinakA prayoga hama Aja kara rahe haiM, vistRta va spaSTa rUpa pradAna kiyaa| vedoM meM jo siddhAMta va vidhiyAM sUtra rUpa meM haiM, ve isa kAla meM apanI pUrNa saMbhAvanAoM ke sAtha jana sAdhAraNa ke sAmane aaiiN| bhAratIya gaNita ke isa svarNayuga kI smRti meM bhArata ne aMtarikSa meM jo prathama upagraha sthApita kiyA usakA nAma AryabhaTTa ke nAma para rakhA gyaa| isa kAla ke katipaya mahAn gaNitajJoM evaM unakI kRtiyoM kA saMkSipta varNana agravat isameM jyA (sine), utkrama jyA (Verse-Sine) tathA koTijyA (Cosine) ke mAna kA ullekha hai| dhyAna rahe ki yahI jyA va jIva araba jAkara jeba (Jaib) ho gayI jisakA zabdika anuvAda laiTina meM jeba (pAkiTa) arthAt Sinus kiyA gyaa| yahI Sine zabda Age calakara Sinus ho gyaa| isI prakAra koTijyA (Cosine) ho gyaa| isa prakAra spaSTa hai ki trikoNamiti ke vistAra meM bhAratIyoM kA bahumUlya yogadAna rahA hai| trikoNamiti zabda zuddha bhAratIya hai jo kAlAntara meM Trigonometry ho gyaa| bhAratIyoM ne trikoNamiti kA prayoga grahoM kI sthiti, gati Adi ke nirdhAraNa meM kiyaa| isa kAla meM bIjagaNita kA vistAra gaNita ke kSetra meM krAntikArI parivartana thaa| vakSAlI pANDulipi meM iSTa karma (Rule of false position) meN| athavA 100 ke sthAna para avyakta rAzi kalpita kI gayI hai| gaNitajJoM kI mAnyatA hai ki iSTakarma hI bIjagaNita ke vistAra kA Adi srota hai| bhAratIyoM ne dhana, RNa jo cinha mAtra haiM, unake jor3a, ghaTAnA, guNA, bhAga Adi ke lie niyamoM ko vikasita kiyaa| mahAn gaNitajJa brahmagupta (628 I0pU0) ne batAyA ki "dhana saMkhyA aura RNa saMkhyA kA guNanaphala RNa saMkhyA, do RNa saMkhyAoM kA guNanaphala dhana saMkhyA tathA dhanasaMkhyA aura dhanasaMkhyA kA guNanaphala dhana saMkhyA hotI hai|" bhAratIyoM ne varga, dhana Adi ghAtoM (Powers) ke lie saMketoM kA prayoga kiyaa| vahI saMketa Aja bhI prayukta kiye jAte haiN| anuyoga dvAra sUtra meM kucha bIjagaNitIya ghAta niyama diye hue haiN| jaise ka kA prathama varga - ka ka kA dvitIya varga = (ka) - ka. ka kA tRtIya varga = (ka) = ka ka kA prathama mUla = ka = ka 1/2 ka kA dvitIya mUla = ka = ka 1/4 aadi| ni:saMdeha aMkagaNita kI bhAMti bIja gaNita bhI bhArata se araba phuNcaa| araba deza ke gaNitajJa (AL-KHOWARIZMI) apanI pustaka "alajabra" va "ala mukAbalA' meM bhAratIya bIjagaNita para AdhArita viSaya kA pratipAdana kiyA hai| unakI pustakoM ke nAma para isa viSaya kA nAma (Algebra) par3a gyaa| jahAM taka anya rASTroM kI bAta hai, hama pAte haiM ki "yUnAnI gaNita" ke svarNayuga meM Adhunika artha meM alajabarA kA nAmonizAna taka na thaa| Classical Period meM yUnAnI loga bIjagaNita ke aneka kaThina praznoM ko hala karane kI yogyatA rakhate the, parantu unake sabhI hala jyAmitIya hote the| bIjagaNitIya hala vahAM sarvaprathama Diaphantus (275 I0 lagabhaga) ke granthoM meM pAye gaye haiN| usa samaya bhAratIya loga bIja gaNita meM anya rASTroM se bahuta Age the| madhyakAla yA svarNayuga (400 I0 se 1200 I0 taka): isa kAla ko bhAratIya gaNita kA svarNayuga kahA jAtA hai, kyoMki isa kAla meM AryabhaTTa, brahmagupta, mahAvIrAcArya, bhAskarAcArya jaise aneka zreSTha evaM mahAn gaNitajJa hue jinhoMne gaNita kI sabhI zAkhAoM ko, AryabhaTTa (499 I0 - prathama): ye paTanA ke nivAsI the| AryabhaTTa ne apanI pustaka 'AryabhaTTIya gaNitapAda' ke 332 zlokoM meM gaNita ke mahatvapUrNa evaM mUlabhUta siddhAMtoM ko sArarUpa meM kaha diyA hai| AryabhaTTIya ke prathama do pAdoM meM gaNita tathA aMtima do pAdoM meM jyotiSa kA vivaraNa hai| prathama pAda dazamItikA meM bar3I-bar3I saMkhyAoM ko varNamAlA ke akSaroM dvArA nirUpita karane kI vidhi bhI batAI gayI hai| isake dvitIya pAda "gaNita pAda" meM aMkagaNita, rekhAgaNita, bIjagaNita aura trikoNamiti ke aneka kaThina praznoM kA samAveza hai| bIjagaNita meM sAdhAraNa tathA vargIya samIkaraNa kuTaka arthAt anirNIta samIkaraNa kA vistRta vivaraNa hai| AryabhaTTa ne trikoNamiti meM sarvaprathama vyutkrama jyA kA prayoga kiyA jo bAda meM pAzcAtya jagata meM (verse sine) ke nAma se jAnA jAne lgaa| rekhA gaNita ke kSetra meM inhoMne IT kA mAna dazamalava ke cAra sthAnoM taka 3.1416 jJAta kiyA jo Aja bhI sahI mAnA jAtA hai| inhoMne batAyA ki 20,000 ikAI vyAsa vAle vRtta kI paridhi kA mAna 62,832 ikAI hotA hai arthAt... T62832/20,000 = 3.1416 aMkagaNita ke kSetra meM bhI inhoMne vargamUla evaM ghanamUla jJAta karane kI vidhiyoM evaM trairAzika niyama kA bhI ullekha kiyA hai| yaha harSa kI bAta hai ki sabase pahale Arya bhaTTa ne hI spaSTa rUpa se pratipAdita kiyA ki pRthvI gatizIla aura sUrya sthira hai| jise pazcima meM koparanikasa ne 1100 varSoM bAda 16vIM zatAbdI meM svIkAra kiyA aura 1642 meM gailIliyo ko isI bAta para aMdhA kiyA gyaa| varAhamihira (550 I): ye AryabhaTTa ke bAda mahAn gaNitajJa evaM dArzanika the| inake grantha paMca siddhAntikA, vRhatsaMhitA, vRhajjAtaka haiN| varAhamihira dvArA racita 'vArAhazulvasUtra' inakI prasiddha kRti hai| isameM gaNita ke sabhI aMgoM para samucita prakAza DAlA gayA hai| vediyoM ke nirmANa saMbaMdhI bIja ke bahuta se sUtra isameM haiN| bhAskara (prathama) (600 I0) : bhAskara ne apanI pustaka mahAbhAskarIya, AryabhaTTIya bhASya aura laghu bhAskarIya meM AryabhaTTa dvArA pratipAdita siddhAMtoM ko aura vikasita evaM vistRta kiyaa| kuTaka (anirdhArya) samIkaraNa para inhoMne ullekhanIya kArya kiyA hai| brahmagupta (628 I0): hIraka jayantI smArikA adhyApaka khaNDa /3
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________________ brahmagupta ne 'bAhmasphuTa siddhAMta' ke 25 adhyAyoM meM se 2 adhyAyoM meM gaNitIya siddhAMtoM evaM vidhiyoM kA vistRta varNana kiyaa| unhoMne gaNita kI 20 kriyAoM tathA 8 vyavahAroM para prakAza ddaalaa| bIjagaNita meM samIkaraNa-sAdhanoM ke niyamoM kA ullekha kiyA tathA anirNIta dvighAtIya samIkaraNa kA samAdhAna bhI btaayaa| jise Ayalara ne 1764 I0 meM aura lAMgreja ne 1768 I0 meM pratipAdita kiyaa| brahmagupta ne samakhAta (Prisn) aura sUcI (Cone and conical column)ke ghanaphala jJAta karane kI vidhi jJAta kii| brahmagupta ne guNottara zreNI kA yoga jJAta karane kI vidhi aura samakoNa tribhuja ke zulva sUtra kA vistRta varNana bhI kiyA hai| brahmagupta ne sarvaprathama ananta kI kalpanA kI aura batAyA ki koI bhI RNa athavA dhanarAzi zUnya se vibhAjita hone para ananta ho jAtI hai| mahAvIrAcArya (850 I0): mahAvIrAcArya jaina dharmAvalambI the| ye amoghavarSa-nRpatuMga nAmaka rASTrakUTa nareza ke sabhAsada the| amoghavarSa nRpatuMga maisUra tathA anya kannar3a kSetroM ke zAsaka the tathA navIM zatI ke pUrvArddha meM siMhAsanArUr3ha hue the| ata: anumAna lagAyA jAtA hai ki mahAvIrAcArya 850 I0 ke lagabhaga hue hoNge| mahAvIra ne 'gaNitasAra saMgraha' nAmaka aMkagaNita ke vRhat grantha kI racanA kii| isameM aMkagaNita kA vidhivat varNana kiyA gayA hai tathA usameM saMgrahIta prazna atyanta manoraMjaka haiN| unhoMne la0sa0a0 kA Adhunika niyama jJAta kiyA jisakA yUropa meM pahalI bAra prayoga 1500 I0 meM kiyA gyaa| inhoMne vRttAntargata caturbhuja tathA dIrgha vRtta ke kSetraphala jJAta karane ke sUtroM kA nirgamana bhI kiyaa| zrIdharAcArya (850 I0): zrIdharAcArya ne aMkagaNita para navazatikA tathA trizatikA, pATIgaNita aura bIjagaNita pustakoM kI racanA kii| inakI racanA zailI atyanta sarala, saMkSipta tathA hRdayagrAhI hai| ataeva ina granthoM kA atyadhika pracAra huA aura sampUrNa bhAratavarSa meM inakA paThana-pAThana huaa| inakI bIjagaNita kI pustaka aprApya hai kintu inakA dvighAta samIkaraNa hala karane kA sUtra jo "zrIdharAcArya vidhi" kahalAtA hai, Aja bhI vyApaka rUpa se vyavahRta ho rahA hai| inakI pustaka "pATI gaNita" kA anuvAda araba meM "hisAbula tarabta" nAma se huaa| AryabhaTTa dvitIya (950 I0): Arya bhaTTa (dvitIya) mahArASTra ke nivAsI the| unhoMne mahAsiddhAMta nAmaka eka grantha likhA jisake eka adhyAya meM aMkagaNita tathA dUsare adhyAya meM prathama ghAta vAle atirdhArya samIkaraNa (kUTaka) kA pratipAdana kiyaa| gole ke pRSThaphala kA zuddha mAna kadAcit isI grantha meM milatA hai| isa grantha meM TT kA mAna 22/7 liyA gayA hai jo Aja bhI sarvamAnya hai| zrIpati mizra (1039 I0): zrIpati mizra bhI mahArASTra ke nivAsI the| inhoMne "siddhAntazekhara" evaM "gaNitatilaka" kI racanA kii| inhoMne kramacaya (Permutation) aura saMcaya (Combination) para vizeSa kArya kiyaa| inakI pustaka gaNita tilaka kA prArambhika bhAga hI prApta ho sakA hai, zeSa bhAga aprApya nemIcandra siddhAnta cakravartI (11vIM zatI): . cakravartI kI prasiddha pustaka 'gommaTasAra' hai jisake do bhAga haiM, karmakANDa evaM jIva kaanndd| donoM meM jIvarAziyoM, unake karmoM, AstavabaMdha, nirjarA, sArvatrika samuccayoM, ekaikI saMgati, sakramabaddhI prameya (well ordering theorems) Adi kI carcA hai| ekaikI saMgati vidhi kA prayoga videzoM meM gailIliyo evaM jArja kaiNTara (1845-1918) dvArA kaI zatAbdiyoM bAda kiyA gyaa| bhAskarAcArya dvitIya (1114 I0): madhyayuga ke antima tathA advitIya gaNitajJa bhAskarAcArya dvitIya ne apanI prasiddha pustaka "siddhAnta-ziromaNi", "lIlAvatI", "bIjagaNita", "golAdhyAya", "gRhagaNitam" evaM "karaNa kutUhala" meM gaNita kI vibhinna zAkhAoM aMkagaNita, bIjagaNita, trikoNamiti Adi ko eka prakAra se aMtima rUpa diyA gayA hai| vedoM meM jo siddhAMta sUtra rUpa meM the, unakI pUrNa abhivyakti bhAskarAcArya kI racanA meM huI hai| inameM brahmagupta dvArA batAyI gaI 20 prakriyAoM aura 8 vyavahAroM kA alaga-alaga vivaraNa aura unameM prayoga meM lAyI jAne vAlI vidhiyoM kA pratipAdana suvyavasthita aura susAdhya rUpa meM kiyA gayA hai| ___ lIlAvatI meM saMkhyA paddhati kA jo AdhArabhUta aura sRjanAtmaka pratipAdana kiyA gayA hai, vaha Adhunika aMkagaNita aura bIjagaNita kI rIr3ha hai| lIlAvatI ke kAvyapUrNa sarala prazna Aja bhI kalA aura vijJAna ke saMgama ke anUThe udAharaNa haiN| bhAskarAcArya dvitIya dvArA bIjagaNita ke anirNIta samIkaraNoM ke samAdhAna ke lie prayukta cakravAta vidhi kI prasiddha vidvAna heMkala ne bhUri-bhUri prazaMsA karate hue kahA hai, "yaha nizcaya rUpa se saMkhyA siddhAMta meM lAyAMja se pUrva sarvotkRSTa upalabdhi hai|" isI niyama kA pharamaTa ne 1657 meM prayoga kiyaa| ___ apanI pustaka "siddhAnta ziromaNi" meM bhAskarAcArya dvitIya ne trikoNamiti kA vistRta ullekha kiyA hai| jyA, kojyA, utkramajyA ke vibhinna sambandha aura tAlikA, avakalana, aNucalana-kalana (Infiritesimal Calcules) tathA samAkalana (Integration) kA bhI pratipAdana kiyaa| pRthvI kI gurutvAkarSaNa zakti kI bhI pUrI jAnakArI unheM thii| unhoMne likhA hai ki pRthvI meM AkarSaNa zakti hai jo vastuoM ko khIMcakara apane dharAtala para lAtI hai| uttara madhyakAla (1200 I0 se 1800 I0 taka): bhAskarAcArya dvitIya ke bAda gaNita meM maulika kArya adhika nahIM ho skaa| prAcIna granthoM para TIkAeM hI uttara madhyakAla kI mukhya dena haiN| kerala ke gaNitajJa nIlakaNTha ne 1500 I0 meM eka pustaka meM jyA x kA mAna nikAlA: jyA x = x-x/3 + x/5 malayAlama pANDulekha "muktibhAsa" meM bhI yaha sUtra diyA gayA hai| jise Aja hama gregarI zreNI ke nAma se jAnate haiN| isa kAla ke gaNitajJoM evaM unakI kRtiyoM kA vivaraNa agrAMkita hIraka jayantI smArikA adhyApaka khaNDa/4
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________________ hai nArAyaNa paNDita (1356 I0): nArAyaNa paNDita ne aMkagaNita para "gaNitakaumudI' nAmaka eka vRhad graMtha kI racanA kii| isameM aneka viSayoM kA pratipAdana kiyA gayA hai| mAyAvarga (Magic Squares) pramukha haiN| yaha graMtha aitihAsika dRSTi se atyanta mahatvapUrNa hai| nIlakaNTha (1587): nIlakaNTha ne "tAjikanIlakaNThI" nAmaka grantha kI racanA kI jisameM jyotiSa gaNita kA pratipAdana kiyA gayA hai| kamalAkara (1608 I0): kamalAkara ne 'siddhAnta tatva viveka' nAmaka graMtha kI racanA kii| samrATa jagannAtha (1731 I0): samrATa jagannAtha ne "samrATa siddhAnta" tathA "rekhAgaNita" nAma kI do pustakeM likhiiN| rekhA-gaNita kI vartamAna zabdAvalI adhikAMzata: isI pustaka para AdhArita hai| ___ uparyukta ke atirikta isa kAla meM keralIya gaNitajJoM meM saMgama grAma meM janme mAdhava (1350-1410) 'yuktibhASA' nAmaka gaNita graMtha ke lekhaka jyeSTha deva (1500-1610 I0) tathA 'kriyAkramakarI' nAmaka lIlAvatI vyAkhyA ke racayitA zaMkara pArazava (1500-1560 I0) ke nAma vizeSa ullekhanIya sudhAkara dvivedI: sudhAkara dvivedI ne 'dIrgha vRtta lakSaNa', 'golIya rekhA gaNita', 'samIkaraNa mImAMsA', 'calana kalana' Adi aneka pustakoM kI racanA kii| sAtha hI brahmagupta evaM bhAskara kI pustakoM para TIkAeM likhakara sAmAnya janatA ke lie sulabha kraayaa| rAmAnujam (1889 I0): __ sUtra meM gaNitIya evaM anya siddhAMtoM ko likhane evaM siddha karane kI vaidika paramparA ke Adhunika yuga ke mahAn gaNitajJa rAmAnujam haiN| unakI zreSThatA isI se pramANita hai ki unake dvArA pratipAdita 50 prameyoM meM se eka do ko siddha karane meM hI gaNitajJoM evaM zodhakartAoM ko varSa paryanta dattacitta hokara parizrama karanA pdd'aa| kucha prameya abhI taka siddha nahIM kiye jA sake haiN| unakI kRti 'rAmAnujam DAyarI' zIrSaka se janmazatI varSa meM prakAzita huI hai| svAmI bhAratI kRSNatIrthajI mahArAja (1884-1960 I0): ___ mahAn gaNitajJa evaM dArzanika jagadguru zaMkarAcArya bhAratI kRSNatIrthajI Adhunika yuga meM vaidika gaNita ke pradhAna bhASyakAra haiN| unhoMne apanI pustaka "vaidika gaNita" meM vaidika sUtroM ko punaH pratipAdita kiyA hai aura unameM nihita siddhAMtoM aura vidhiyoM ko itanI sarala, sugrAhya evaM suspaSTa bhASA meM prastuta kiyA hai ki gaNita kA eka sAdhAraNa vidyArthI bhI use AtmasAt kara gaNita ke jaTilatama praznoM ko atyalpa samaya meM hala kara sakatA hai| unakI pustaka vaidika gaNita para eka prAmANika graMtha hai| svAmIjI ne apanI isa anupama kRti ke mAdhyama se vaidika gaNita meM chipI huI adbhuta kSamatA se paricaya karAkara gaNita ke kevala sAmAnya vidyArthI ko hI nahIM apitu adhikArI vidvAnoM ke anta:karaNa ko bhI jhaMkRta kara diyA hai| inhoMne hameM sarvathA navIna dRSTi dekara vaidika gaNita para zodha karane tathA usakA upayoga karane ke lie vivaza kara diyA hai| saha zikSaka-zrI jaina vidyAlaya, kalakattA vartamAna kAla (1800 I0 ke pazcAt): vartamAna kAla ke kucha pramukha gaNitajJoM evaM unakI kRtiyoM kA saMkSipta paricaya nimnavat hainRsiMga bApU deva zAstrI (1831 I0) : nRsiMga bApU deva zAstrI ne bhAratIya evaM pAzcAtya gaNita para pustakoM kA sRjana kiyaa| inakI pustakoM meM rekhAgaNita, trikoNamiti, sAyanavAda hIraka jayantI smArikA adhyApaka khaNDa /5